Since \(\tilde{P}\)  is almost surely discrete, if one wishes to model continuous data one may convolve it with a density kernel \(k\)  parametrized by a latent variable \(\theta \)  that is drawn from a Dirichlet process. This yields the popular Dirichlet process mixture [1]}, which exhibits appealing asymptotic properties in the context of density estimation: in several relevant cases, the posterior distribution concentrates at the true data-generating density at the minimax-optimal rate, up to a logarithmic factor, as the sample size increases [2]}, [3]}.
Such a model and mevery of its variants are widely used across scientific areas, thanks also to the availability of a wide variety of efficient computational methods to perform inference, see for instance [4]}, [5]}, [6]}, [7]}, [8]}.
